The
main goals of this book are to provide an overview
of various mathematical, statistical and
computational methodologies used in computational
neuroanatomy to a wide range of researchers and
students, and to address important yet technically
challenging topics in further detail.

We present a
novel cosine series representation for encoding
fiber bundles consisting of multiple 3D curves using
the cosine series expansion. We address the issue of
registration, averaging and statistical inference on
curves in a unified Hilbert space framework. MATLAB

We present a
novel computational framework for characterizing
signal in brain images via nonlinear pairing of
critical values of the signal. Among the
astronomically large number of different pairings
possible, we show that representations derived from
specific pairing schemes provide concise
representations of the image.

We
present a novel framework for characterizing
signals in images using techniques from
computational algebraic topology. The main tool
is persistent homology which can be encoded in
persistence diagrams. These diagrams
visually show how the number of connected
components of the sublevel sets of the signal
changes.

Most previous
approaches start with flipping the 3D magnetic
resonance images (MRI). The anatomical
correspondence across the hemispheres is then
established by registering the original image to the
flipped image. We present a radically different
asymmetry analysis that utilizes a novel weighted
spherical harmonic representation of cortical
surfaces.

Chung,
M.K., Dalton, K.M., Davidson, R.J. 2008. . Tensor-based
cortical surface morphometry via weighed spherical
harmonic representation. IEEE Transactions on
Medical Imaging27:1143-1151 (invited submission based on MMBIA
2006 oral presentation). MATLABWe present a
new tensor-based morphometric framework that
quantifies cortical shape variations using a local
area element. The local area element is computed
from the Riemannian metric tensors, which are
obtained from the smooth functional parametrization
of a cortical mesh. For the smooth parametrization,
we have developed a novel weighted spherical
harmonic (SPHARM) representation.

We present a
novel data smoothing and analysis framework for
cortical thickness data defined on the brain
cortical manifold. When the observations lie
on a convoluted brain surface, however, it is more
natural to assign the weights based on the geodesic
distance along the surface. We therefore develop a
framework for geodesic distance-based kernel
smoothing and statistical analysis on the cortical
manifolds.MATLAB